#6806835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,28)(13,19)(14,18)(15,21)(16,25)(17,32)(20,30)(23,34)(24,35)(26,33)(27,38)(29,36)(31,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,16,18,22)(12,29,31,13)(15,26,30,27)(17,34,35,39)(19,37,36,28)(20,33,21,38)(23,32,40,24);(9,12,30,32,14,31,15,24)(10,17,20,37,18,35,21,28)(11,23,26,13,25,40,27,29)(16,19,38,39,22,36,33,34);(2,6)(5,7)(10,18)(16,22)(17,35)(19,36)(20,21)(28,37)(33,38)(34,39);(3,8)(5,7)(9,13,15,23,14,29,30,40)(10,19,21,34,18,36,20,39)(11,24,27,31,25,32,26,12)(16,17,33,28,22,35,38,37)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714031);